A Diffusion Equation with a Nonmonotone Constitutive Function

Höllig, Klaus; Nohel, John A.

doi:10.1007/978-94-009-7189-9_26

Cited by 28 publications

(23 citation statements)

References 2 publications

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“…(2.10). More definitive statements about the solutions of the PDE require careful analysis [14,20]. In the following sections, we study in detail the stability, local instabilities and global dynamics of the discretized model (2.25).…”

Section: Comparison Of Discrete Equilibria With Generalized Solutionsmentioning

confidence: 99%

A discrete model for an ill-posed nonlinear parabolic PDE

Witelski

Schaeffer

Shearer

2001

Physica D: Nonlinear Phenomena

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We study a finite-difference discretization of an ill-posed nonlinear parabolic partial differential equation. The PDE is the one-dimensional version of a simplified two-dimensional model for the formation of shear bands via anti-plane shear of a granular medium. For the discretized initial value problem, we derive analytically, and observed numerically, a two-stage evolution leading to a steady-state: (i) an initial growth of grid-scale instabilities, and (ii) coarsening dynamics. Elaborating the second phase, at any fixed time the solution has a piecewise linear profile with a finite number of shear bands. In this coarsening phase, one shear band after another collapses until a steady-state with just one jump discontinuity is achieved. The amplitude of this steady-state shear band is derived analytically, but due to the ill-posedness of the underlying problem, its position exhibits sensitive dependence. Analyzing data from the simulations, we observe that the number of shear bands at time t decays like t −1/3 . From this scaling law, we show that the time-scale of the coarsening phase in the evolution of this model for granular media critically depends on the discreteness of the model. Our analysis also has implications to related ill-posed nonlinear PDEs for the one-dimensional Perona-Malik equation in image processing and to models for clustering instabilities in granular materials. © 2001 Elsevier Science B.V. All rights reserved.Keywords: Nonlinear PDE; Ill-posed equations; Nonlinear diffusion; Granular medium; Shear bands BackgroundThe degenerate parabolic PDE:in which R is a rotation matrix, arises from a simplified model of the velocity field of a sheared granular material [33]. This equation is ill-posed, a property typical of continuum models of granular media [25,31,32,34]. To study the dynamics of this model, in this paper, we analyze one particular finite-difference approximation of the PDE (1.1).More precisely, we discretize in space, and study the resulting system of ODEs, which we refer to as the discrete model. This model is a form of regularization of Eq. (1.1); the discrete model is well-posed, mollifying instabilities at the highest frequencies.

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Section: Comparison Of Discrete Equilibria With Generalized Solutionsmentioning

confidence: 99%

A discrete model for an ill-posed nonlinear parabolic PDE

Witelski

Schaeffer

Shearer

2001

Physica D: Nonlinear Phenomena

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“…T(x -|, /)/"(*) di-\Tx{x-s(r), t -t)v((s(t), t) dr. where (see (6) and (7)) /•(?) = df(s(t))/dt = ux(.?…”

Section: R(t)=f(s(t))-f(0)mentioning

confidence: 99%

“…The present study of (1), (2) is intended as a step towards the understanding of this intriguing phenomenon. The relation of the convexified problem in [6] to the Cauchy problem (1), (2) is clear (the particular boundary conditions in [6] do not play a role in the analysis of the free boundary curve).…”

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A nonlinear integral equation occurring in a singular free boundary problem

Höllig¹,

Nohel²

1984

Trans. Amer. Math. Soc.

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Abstract. We study the Cauchy problem j u, = (ux)x, (x, ?) e R x R + , \"(,0)=/ with the piecewise linear constitutive function (|) = £+ = max(0, |) and with smooth initial data/which satisfy xf'(x) > 0, x e R, and/"(0) > 0. We prove that free boundary s, given by ux(s(t)+, t) = 0, is of the formwhere the constant k = 0.9034... is the (numerical) solution of a particular nonlinear equation. Moreover, we show that for any a e (0,1/2), /W'))=o('°"'). '-o+.The proof involves the analysis of a nonlinear singular integral equation.

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“…In [4] the well-posedness of the initial-boundary value problem for (0.1) on D = [0,1] X [0,7] is discussed in the context of homogeneous Neumann boundary conditions. An especially interesting result in [4] is a maximum-minimum principle which, among other things, yields the result that <t>'(ux) is positive on D provided that <b'(f) is positive on [0,1] where/is the initial temperature distribution, and u is a smooth (W2'2) solution of (0.1). Thus if /is smooth and <?>'(/') is positive on [0,1], then the initial-boundary value problem for (0.1) can be treated as an ordinary (i.e.…”

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confidence: 99%

“…The specific examples of tb analyzed earlier in [3,5,6] have this property and indeed satisfy all of the above restrictions. Some of the analysis in [4] deals with specific examples that also satisfy them although the a priori estimates included in [4] would apply to more general forms of <b.…”

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confidence: 99%